Maximal heat dissipation capacity and hyperthermia risk: neglected key factors in the ecology of endotherms


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1. The role of energy in ecological processes has hitherto been considered primarily from the standpoint that energy supply is limited. That is, traditional resource-based ecological and evolutionary theories and the recent ‘metabolic theory of ecology’ (MTE) all assume that energetic constraints operate on the supply side of the energy balance equation.

2. For endothermic animals, we provide evidence suggesting that an upper boundary on total energy expenditure is imposed by the maximal capacity to dissipate body heat and therefore avoid the detrimental consequences of hyperthermia – the heat dissipation limit (HDL) theory. We contend that the HDL is a major constraint operating on the expenditure side of the energy balance equation, and that processes that generate heat compete and trade-off within a total boundary defined by heat dissipation capacity, rather than competing for limited energy supply.

3. The HDL theory predicts that daily energy expenditure should scale in relation to body mass (Mb) with an exponent of about 0·63. This contrasts the prediction of the MTE of an exponent of 0·75.

4. We compiled empirical data on field metabolic rate (FMR) measured by the doubly-labelled water method, and found that they scale to Mb with exponents of 0·647 in mammals and 0·658 in birds, not significantly different from the HDL prediction (> 0·05) but lower than predicted by the MTE (< 0·001). The same statistical result was obtained using phylogenetically independent contrasts analysis. Quantitative predictions of the model matched the empirical data for both mammals and birds. There was no indication of curvature in the relationship between Loge FMR and LogeMb.

5. Together, these data provide strong support for the HDL theory and allow us to reject the MTE, at least when applied to endothermic animals.

6. The HDL theory provides a novel conceptual framework that demands a reframing of our views of the interplay between energy and the environment in endothermic animals, and provides many new interpretations of ecological and evolutionary phenomena.

‘...being conservative is a bigger problem than being too speculative.

Brian K. McNab (2002)


The second law of thermodynamics demands that organisms comply with an energy balance equation where energy input is balanced by energy output (plus or minus storage)

image(eqn 1)

Most considerations of energy, in the context of ecology, have focussed on the role of its supply (energy input) as a potentially limiting factor on survival and reproduction (see for example Stearns 1976, 1992; McNab 2002). Accordingly, energy is regarded as a limited resource that must be harvested from the environment before it can be allocated among competing demands (energy outputs). This view of energy dates back at least to the 1930s and the modern evolutionary synthesis (Huxley 1942), as exemplified by the classic resource allocation problem identified by Fisher (1930). The competing demands for energy output include metabolic processes (like basal metabolism and immune function) and various behaviours (such as foraging and escaping from predators) that together ensure survival. Alternatively, animals may invest resources in reproduction, which is also energetically costly (e.g. Kaczmarski 1966; Migula 1969; Millar 1975, 1978; Loudon & Racey 1987; Kenagy, Stevenson & Masman 1989; Kenagy et al. 1990; Hammond & Diamond 1992, 1994; Rogowitz 1996, 1998; Speakman 2008). Because so far the axiomatic position in ecology is that energy supply (the left hand side of the energy balance equation) in the environment is limited, there must consequently be an optimal allocation of energy between competing processes that require energy (the right hand side of the equation), which then drives the ecology.

This view of energy may in some circumstances be correct. The seasonal timing of reproduction to coincide with annual cycles of primary productivity (Thomas et al. 2001) or the use of strategies such as hibernation (Heldmaier & Klingenspor 2000) and migration (Newton 2008) to avoid seasonal periods of low food supply are among obvious examples. However, it is our fundamental contention that in many situations animals experience intrinsic physiological limitations that supersede limits to energy supply. We suggest that failure to recognise this has been a key flaw in our understanding of the relationship between energy and ecology. We contend that animals spend their lives shuttling between situations where energy supply is limited and situations where they are physiologically constrained, and hence supply is effectively unlimited. This leads to several novel interpretations of existing phenomena and a number of predictions. Ultimately, the recognition that in many circumstances animals are not externally limited by food resources demands a reframing of our wider understanding of many ecological processes.

If energy supply limits reproductive output then supplementation experiments where extra food is provided should lead to increases in reproductive performance. Boutin (1990) reviewed 138 studies of food supplementation in vertebrates (129 of them endotherms). Most of these studies did not restrict supplemental feeding only to the breeding season. The commonest response (33/39 studies) was to advance the date of first breeding. This could be interpreted as moving forwards the transition from when food is limited to when it is not. A second common response was to reduce the home range size (19/23 studies), consistent with feeding only until a fixed energy intake had been achieved. In contrast, only 9/21 studies reported increases in litter or clutch size, and in all cases reported these increases were quite modest, and perhaps explained by the reallocation of the energy that had formerly been utilised to collect energy from a larger home range.

A simple consideration of animal morphology supports the contention that environmental energy supply is often not limiting. Almost all vertebrates have at the entrance to their alimentary tract a storage organ. For example, in mammals, reptiles, fish and amphibians it is the stomach, in birds it is the crop. These structures exist because animals are able to harvest food from their environment at a much faster rate than they can process and utilise it. The storage organ is required to match together the rates at which energy can be collected from the environment with the rates at which it can be used. The universal existence of these structures indicates that for almost all animals, the rate at which energy can be collected from the environment must at some phases of their lives be considerably greater than the rate at which it can be utilised.

The recently articulated ‘metabolic theory of ecology, MTE’ (West, Brown & Enquist 1997, 1999, 2000; Gillooly et al. 2001; Brown et al. 2002, 2004; Savage et al. 2004) is an attempt to understand from first principles how body size and body temperature (Tb) may influence the capacity of an organism to acquire and expend energy. This theory is interesting because it shifts emphasis from a limited energy supply in the environment to a limited energy supply within the organism. The fundamental basis of the MTE is that animals acquire resources and allocate them to their needs via a fractal distribution system (West, Brown & Enquist 1999; Brown et al. 2002, 2004). The physical constraints on how this system scales with body size then influence how animal energy demand and ecological processes scale with body size. Fundamentally, this idea is also based on the premise of a supply side limitation, i.e. the constraint lies on the left hand side of the energy balance equation (eqn 1).

In the current review, we propose that animals are frequently constrained by intrinsic physiological factors that govern their capacity to expend energy, shifting the limit on performance from the left to the right hand side of the energy balance equation. We argue that understanding the factors that limit the capacity of organisms to expend energy may then provide valuable insights into key ecological processes. The suggestion that animals may be constrained by their capacity to expend energy, rather than their capacity to acquire it, has been made before. The ‘peripheral limitation hypothesis’ (Hammond & Diamond 1997; Bacigalupe & Bozinovic 2002) suggests that the sustained rate of energy expenditure by an animal is dependent on the summed maximal metabolic rates of its component tissues and organs. For example, in the context of lactation, the performance of a female and her maximal ability to utilise energy was suggested to be constrained by the capacity of the mammary gland to process ingested energy into milk. The factors that might limit these individual components are unstated, and by this model there is no common constraining factor across different modes of energy use. Hence, in one state the maximal energy flux may be limited by the mammary gland capacity, but in another it might be constrained by capacity of the brown adipose tissue to generate heat, and in yet another by the capacity of skeletal muscle to perform mechanical work. We suggest here that viewing limitation on energy flux as residing on the output side of the energy balance equation, as articulated in the peripheral limitation hypothesis, is correct. Most importantly, we extend this idea by suggesting that there is a common constraining factor that limits maximal energy expenditure independent of the uses to which the energy is allocated. This common factor is the maximum ability to dissipate body heat. We call this idea the heat dissipation limit (HDL) theory. This theory challenges the suggestion that energy supply is limited, either by extrinsic factors linked to the environment or morphological features relating to resource acquisition and distribution (the MTE).

This review consists of four parts. First, we provide a description of the origin of the HDL theory. Second, to test the theory we derive a mathematical model which predicts how maximal heat loss capacity of animals should vary with body size. The prediction from the HDL theory differs substantially from the prediction of the MTE, allowing us to test between these different conceptualisations of how energy and metabolic rate are related to ecology. Third, we have compiled data on the daily rates of energy expenditure in free-living mammals and birds (field metabolic rate, FMR) to test between the different models. This comparison strongly supports the HDL theory over the MTE. Finally, we explore some additional implications and predictions of the HDL theory and discuss some of its potential problems.

Origin of the heat dissipation limit theory

Lactation is widely recognised to be the most energy demanding period in the mammalian life cycle (e.g. Millar 1975, 1978; Oftedal 1984; Loudon & Racey 1987; Gittelman & Thompson 1988; Kenagy, Stevenson & Masman 1989, Kenagy et al. 1990; Hammond & Diamond 1992, 1994; Rogowitz 1996, 1998; Speakman 2008). During lactation, mammals (particularly small ones) massively increase their energy intake to support the production of milk that supplies the energy used by their offspring. The intake of energy increases rapidly in early lactation, but then reaches a maximum (Johnson, Thomson & Speakman 2001a). Experimental manipulations that involve increasing litter size (Hammond & Diamond 1992; Johnson, Thomson & Speakman 2001a), making females simultaneously pregnant (Johnson, Thomson & Speakman 2001b), artificially extending the period of lactation (Hammond & Diamond 1994; Laurien-Kehnen & Trillmich 2003) or forcing the mothers to exercise when lactating (Perrigo 1987) do not stimulate lactating females to increase their energy intake or milk production. This fundamental limit on lactation performance generates the widely observed and ecologically important trade-off between litter size and offspring size (e.g. Williams 1966; Smith & Fretwell 1974; Roff 1992; Stearns 1992), because larger litters must divide between them a fixed milk supply. Interest in the factors that cause this limited energy intake at peak lactation spans at least the last three decades, primarily because this is the highest rate at which mammals ingest food, and hence it potentially indicates a boundary on energy intake to support other metabolic processes.

An early idea was that intake was regulated by the capacity of the alimentary tract to process incoming energy. Hence, energy may be abundant in the environment and ingested rapidly into the alimentary tract, where it is stored in the stomach or crop, but the uptake capacity of the tract itself restricts the level of expenditure – the ‘central limitation hypothesis’ (Drent & Daan 1980; Weiner 1989, 1992; Peterson, Nagy & Diamond 1990; Koteja 1996a). This idea predicts that maximal sustained energy expenditure should be independent of what the animal does because sustained performance is controlled by the fixed intake of energy from the alimentary tract (Weiner 1992). Support for this idea was equivocal (Koteja 1996b), ultimately because the alimentary tract is extraordinarily flexible in structure and uptake capacity in response to changing demands (Hammond & Wunder 1991; Hammond & Diamond 1992; Starck 1999; Naya, Bozinovic & Karasov 2008). In a key experiment dismissing the central limitation hypothesis (Hammond et al. 1994), it was shown that when lactating mice were exposed to the cold, they were able to elevate their food intake above the level that was apparently a limit when manipulations involved changing litter size, lactation duration or simultaneous exercise. This effect was subsequently replicated in other strains of mice and other species (Rogowitz 1996, 1998; Hammond & Kristan 2000; Johnson & Speakman 2001; Zhang & Wang 2007). Attention therefore shifted to metabolic capacity of the organs where energy was being used, notably the mammary tissue, and experiments were performed demonstrating that the mammary glands seemed to be operating at maximal capacity during peak lactation (Hammond, Lloyd & Diamond 1996). Specifically, female mice were surgically manipulated by removing half of their mammary glands. The rationale behind this experiment was that if the capacity of the mammary glands was limited, then when the mammary tissue was halved in size, the remaining tissue would be unable to compensate. However, if the capacity of the tissue was flexible, then it would respond to the reduction in tissue by expanding its capacity. This could be confounded if the problem of the number of pups relative to the number of teats changed, as removing half the mammary tissue also removed half the outlets for milk to pups. Yet, Hammond and colleagues also manipulated pup number so that either pup numbers were reduced by half as well, or maintained at the same level. They found that productivity in the halved glands did not increase, suggesting that the mammary gland was indeed the point at which the system was peripherally limited. This became known as the peripheral limitation hypothesis.

A fundamental prediction of the peripheral limitation hypothesis is that the mammary glands perform maximally at peak lactation independent of any other energetic challenges experienced by the female during lactation, but total energy intake depends on combined energy demands due to all factors, including for example thermoregulation demands when placed in the cold. Milk production should therefore be independent of ambient temperature. Directly contradicting this prediction, it was shown that during cold exposure (8°C), mice not only elevated food intake but also produced more milk and had heavier offspring than they did at room temperature (21°C) (Johnson & Speakman 2001). Similarly, when mice were exposed to thermoneutral conditions (30°C), they ate less food, produced less milk and smaller offspring (Król, Johnson & Speakman 2003; Król and Speakman 2003a; Król and Speakman 2003b). These observations of temperature effects on lactation performance are consistent with a heat dissipation limit constraining total energy expenditure and reproductive output. Accordingly, when lactating animals were placed in the cold, they elevated intake not because of the increased demand for thermoregulation, but rather because the steeper gradient between body and ambient temperature increased their capacity to dissipate body heat. This allowed them to elevate highly exothermic processes such as milk production. Exposing mice to temperatures exceeding room temperature had the opposite effect of reducing their capacity to dissipate heat and forcing them to suppress milk production. Taken alone, the temperature manipulations during lactation support the HDL theory, but can also be potentially explained by thermal effects of the environment on offspring growth capacity and their demand for milk. To eliminate this possibility, the capacity of lactating mice to lose heat independent of effects on their offspring was manipulated by shaving fur from their dorsal surfaces (Król, Murphy & Speakman 2007). Critically, mice so manipulated increased their food intake by 12·0% and their milk production by 15·2%, leading to elevated pup growth by 15·4%, strongly supporting the HDL theory.

In fact, the role of heat as a factor constraining mammalian reproduction had been known for some considerable time before our ‘discovery’ in lactating mice. The context, however, of these previous studies was different and concerned reproductive performance of domesticated livestock (e.g. Ominski et al. 2002; Lacetera et al. 2003; Renaudeau, Noblet & Dourmad 2003; Odongo et al. 2006). The novelty in our work was to discover that a factor of key importance constraining the lactation performance of large 50–500 kg animals like sheep, pigs and cattle, with low surface to volume ratios, was also important in mice weighing only 30–40 g with a surface to volume ratio 20–30 times greater. The commonality of this constraint across such a large range of body masses suggested the potential for heat dissipation capacity to be a fundamentally unifying factor in our understanding of energetic constraints affecting endotherms.

The HDL theory suggests that ability to lose heat is the key process that controls maximal energy expenditure. The heat dissipation capacity, rather than the ability to acquire resources from the environment, provides the boundary condition that leads to trade-offs in animal energy allocation. If animals generate heat at a greater rate than it can be dissipated, the result is an increase in Tb (hyperthermia). Elevated Tb causes immediate physiological problems and if unchecked can be fatal within a relatively short period (Yan et al. 2006). Hyperthermia that is not immediately fatal may cause longer term negative effects that influence survival and reproductive performance. These negative effects include elevated protein damage and induction of heat shock proteins, an increased inflammatory response, apoptosis, hypoxia, DNA damage in germ cells, oxidative stress, mitochondrial dysfunction, liver damage, cerebral ischemia, a disrupted blood brain barrier, embryonic death in pregnant females and numerous other malfunctions (e.g. Skibba et al. 1991; Kregel 2002; Lacetera et al. 2003; Osorio et al. 2003; Cui et al. 2004; Franklin et al. 2005; McAnulty et al. 2005; Yan et al. 2006; Zhao, Fujiwara & Kondo 2006; Chang et al. 2007; Bloomer et al. 2008; Lee et al. 2008; Ozawa et al. 2008; Sakamoto et al. 2008; Haak et al. 2009; Paul, Teng & Saunders 2009). In contrast to the negative effects of mild hyperthermia, mild hypothermia appears to have beneficial effects. For example, mild hypothermia in the brain is neuroprotective (Uemura et al. 2003; Cheng et al. 2008). Moreover, animals that were transgenically manipulated to over express the uncoupling protein 2 in hypercretin neurons experienced local brain heating in the hypothalamus, and to compensate they reduced their Tb by around 0·5°C (Conti et al. 2006). This manipulation led to an extension of life span by 12% in males and 20% in females. Endothermic animals routinely modulate their Tb by cooling themselves down in situations where resources are limited (e.g. hibernation (Heldmaier & Klingenspor 2000) and daily torpor (Rikke et al. 2003; Schubert et al. 2010)), but seldom utilise strategies of increased Tb because the negative effects of hyperthermia normally offset any benefits. An obvious exception is the use of fever in response to infection (reviewed in Zhang et al. 2008).

Heat dissipation capacity and avoidance of hyperthermia are likely to be factors of key importance in the physiology and ecology of endotherms. By this view, the important ecological trade-off between litter size and offspring size, which is classically interpreted as the inability to acquire greater resources from the environment to provision the larger litters, occurs because milk production is exothermic and thus limited by the maximal capacity to dissipate heat. By extension, we suggest that many trade-offs in energy allocation in endotherms may not be governed by limitations in energy supply, but rather by limits on heat dissipation capacity and the risk of hyperthermia. This view requires a fundamental reinterpretation of many areas of ecological energetics.

Heat dissipation capacity: a theoretical model

Maximum heat dissipation capacity

The theoretical maximum rate of heat loss in relation to animal size can be modelled from first principles of heat exchange. Attempts to model heat exchange processes date back over 170 years to the pioneering work of Fourier (1822). Animals living in a complex environment exchange heat at their surfaces primarily by the processes of radiation and convection (Kleiber 1961; Calder 1984; McNab 2002). The body does not produce heat uniformly. Some tissues such as the brain and kidneys have high rates of metabolism, while others such as adipose tissue and bone have relatively low rates (Elia 1992). Moreover, the diversity of tissue types comprising the body vary in thermal conductivity and diffusivity (Touloukian 1964; Bowman, Cravalho & Woods 1975). However, heat is efficiently distributed through the body by the blood (Wissler 1961), and redistribution of heat occurs quite rapidly relative to its loss from the surface. Hence, tissues that are well perfused tend to have similar temperatures independent of their metabolic rates, thermal conductivities and diffusivities. In the 1960s, this observation led to the notion that heat relations of a living body could be effectively modelled by considering it to consist of just two compartments – a central core maintained at a fixed regulated body temperature (Tb) surrounded by a shell in which temperature is not regulated (e.g. Fan, Hsu & Hwang 1971). The shell may comprise non-perfused tissue or body components with low thermal conductivity, such as blubber or the pelage. The primary mechanism of heat transfer through the shell is via conduction, although there is also in some circumstances heat loss via radiation (Wolf & Walsberg 2000). Most resistance to heat loss from the central core is provided by the shell. Heat loss at the surface of the shell must be balanced by heat input at the inner surface of the shell. We can model the maximal heat loss of the organism simply by examining heat flow across the shell.

In general, the flow of heat from the core, across the insulating shell, to the environment depends on four parameters: the surface area of the organism (A), the depth of the insulating layer (d), the thermal conductivity of the insulating layer (k), and the difference between the core temperature (Tb) and the surface temperature of the shell (Ts) (Wyndham & Atkins 1960). The Ts in turn depends on the radiation and convective heat losses occurring at the body surface, which are themselves dependent predominantly on the ambient temperature (Ta), windspeed and incoming levels of solar radiation, linking together the animals capacity to dissipate heat with variation in environmental conditions (Porter & Gates 1969; Calder & King 1974; Wolf & Walsberg 1996, 2000; Walsberg, Tracy & Hoffman 1997). Following Fourier’s laws, the maximal heat loss across the shell (Hs) is equal to

image(eqn 2)

The parameters k, A, d and Tb are all potentially dependent on body size. We will start the modelling process assuming that the traits influencing heat loss have only been affected by the physical consequences of body size, and are not the end products of natural selection. This allows us to predict how we might expect organisms to respond to these physical consequences of size, rather than building the empirical responses of animals into the model from the outset. This failure to adequately separate prediction from assumption has been a major criticism of the MTE (Horn 2004; Kozłowski & Konarzewski 2004, 2005; Brown, West & Enquist 2005; Gillooly et al. 2006). We assume that Tb is independent of body size. Because we are concerned with the maximal capacity to dissipate heat, we also assume that animals of different body sizes are all in environments where the Ta is lower than the upper boundary of the thermoneutral zone (upper critical temperature). We further assume that there is no incoming solar radiation. If these latter two assumptions are violated, then the capacity to dissipate heat would be more constrained and therefore lower than predicted.

Heat loss in a vertebrate organism can be modelled as heat loss from a cylinder or sphere, with additional cylinders for the limbs. This is a classic one-dimensional Fourier heat transfer problem (Holman 1973). The surface area (A) of a sphere increases in relation to the square of its radius (r), while mass of the sphere increases in relation to the volume (4/3 · π · r3); hence A is proportional to mass2/3 ( = mass0·66). If the body is modelled as a cylinder in which the length of the cylinder changes in direct relation to the radius, then the same relation between surface area and volume pertains, but the constants are different. Heat loss will also depend on the length and radius of the limbs. Several theoretical arguments have been presented concerning how limbs should scale with organism body mass (Mb). These depend in part on ideas of compression strength, buckling and bending for struts required to support a body of a given weight in Earth’s gravity (Rashevsky 1960; McMahon 1973; McMahon 1975a; McMahon 1975b). The theory of elastic similarity predicts that leg length should scale as Mb0·25. As animals get larger, their limb lengths should increase at a relatively slower rate than their other body dimensions. By elastic similarity, limb diameters should increase more rapidly (Mb0·38). Bigger animals should therefore have shorter stockier legs to support their mass. In contrast, arguments of geometric similarity suggest the limbs should increase in length and diameter at the same rate as other linear body dimensions (Mb0·33). Following the arguments of geometric similarity, the total animal surface area increases in relation to Mb0·66, while arguments of elastic similarity predict that surface area should increase in relation to Mb0·63 (McMahon 1973). We follow the common assumption that any internal areas such as the lungs, which have massively enlarged surfaces for gas exchange, do not participate in heat exchange with the environment as heat is normally conserved in the process of conserving water.

We will assume that organisms invest a proportion of their Mb into the shell, and that the shell essentially consists of the pelage. The depth of the pelage can then be estimated from the different radii that span the core body and the core plus shell (Fig. 1). The volume of the core is (4/3 ·π·rc3), where rc is the radius of the core. The volume of the whole body is (4/3 ·π·rt3), where rt is the radius of the core plus shell. The depth of the insulating shell (d) therefore equals rt − rc. If the investment in the pelage is a proportion of Mb (called pp), then the volume of the core body Vc = (1 − pp) ·Mb/ρc, and the volume of the whole body, assuming that core and shell have equal density (mass per unit volume, ρc), Vt = Mb/ρc. Hence,








Therefore, as rt − rc

image(eqn 3)
Figure 1.

 The core and shell model for heat transfer in an endotherm. The central core is maintained at body temperature (Tb). Heat is dissipated (wavy arrows) from the core across the shell by the process of conduction, and lost at the surface by radiation and convection, which depend among other parameters on surface temperature (Ts), ambient temperature (Ta), windspeed and incoming solar radiation. Radius of the core (rc), depth of the shell (d) and total radius of the core and shell (rt) are also shown.

Consequently, d is proportional to Mb0·33. This model makes two critical assumptions. First, the core and shell are both centred on a single point. We will explore the consequences of relaxing this assumption later. Second, the core and shell have the same density. If the shell and core do not have equal densities, then Vc = (1 − pp) · Mb/ρc and Vt = ((1 − pp) ·Mb/ρc + pp·Mb/ρs), where ρc and ρs are the densities of the core and shell, respectively. Hence,





image(eqn 4)

Thus, d will still be proportional to Mb0·33 and inversely proportional to ρs. Mammals and birds can modulate the depth of their pelage using small muscles to erect or flatten the fur or feathers. Hence, the maximum thickness of the pelage occurs when the fur or feathers are vertical relative to the body surface, and the minimum thickness occurs when they are lying flat at the lowest angle relative to the surface. If the maximum and minimum angle are independent of Mb, then the scaling of d in relation to Mb is unaffected (Calder 1984), but the absolute values differ.

Modelling heat transfer through the shell

Modelling the dependence of the thermal conductivity of the shell (k) on body size is complex because in most animals the pelage is not a homogenous structure. Fur and feathers trap within their structure small air cells. Heat is conducted along the shafts of the hair follicles or feathers (Wolf & Walsberg 2000). This process depends on the length of these structures, their cross-sectional area and their thermal conductivity. Heat also passes via the air cells by conduction. Air is a better insulator than either fur or feathers by about a factor of six (Bowman, Cravalho & Woods 1975). This contribution of air to total conductance of the pelage has been neatly shown by measuring thermal conductance of pelts in air and in a vacuum (Walsberg 1988a). In three small bird species, the air contributed about half of the total thermal conductance. Animals might modulate heat flow by altering the relative contribution of the hairs/feathers and air spaces. However, the exact consequences of such morphological strategies become difficult to predict because as the pelage contribution to the shell becomes reduced and the sizes of the air cells increases, then heat starts to be transferred also due to currents in the air cells generated by free convection and by radiation. In the bird species studied by Walsberg (1988a), the heat transfer in air spaces in the pelage was greater by 26–41% than the theoretical conductance of completely still air, demonstrating the importance of free convection in the pelage. If the animal invests a fixed proportion of its Mb into the shell (pp), eqn 4 indicates that there is a trade-off between the density of the shell (ρs) and its depth (d). Having a pelage with a very low density would enable a deep insulative layer retarding heat flow along the shafts of the fur/feathers, and a greater proportional contribution of air. But this would lead to increased heat flow via the larger air cells. In contrast, a dense pelage (high ρs) would not be so deep (low d) and have a lower contribution of air to the total structure, but this air would be more stationary preventing convective losses, and the fur/feather canopy more closed, preventing radiation loss. In essence, these arguments emphasise that the parameter k (thermal conductivity) in eqn 2 is not independent of the parameter d (depth of the pelage).

Another complicating factor is that so far we have assumed that hairs and feathers are structures that are independent of animal size, i.e. of constant radius. It seems logical, however, to assume that hairs follow the same sort of geometric relations to body size as other linear features. Thus, as animals get bigger, we might expect the radii of their hairs to increase in relation to Mb0·33 and their cross-sectional areas to increase in proportion to Mb0·66. At a fixed density of pelage ρs, there would consequently be fewer actual hairs of greater diameter as animals get larger, separated by greater spaces in which free convection would be more likely to occur. In fact, across a range of mammals thermal conductivity is positively related to fur thickness (Birkebak 1966), because larger animals that have thicker fur also have coarser hair fibres. Flattening the pelage further complicates the issue because this alters the density of the pelage through changes in the relative proportion of air and fur/feathers, and modifies the shape of the air cells altering the potential for free convection. Moreover, overlapping hairs/feathers provide a greater target for interception of radiation. This will retard loss of radiation from the surface, but also increase uptake of incoming radiation, which we have excluded in our model, but which in practice may be an important issue for animals in the wild. In pigeons Columba livia, empirical measurements indicated that completely flattening the plumage reduced its depth from 31 to 8 mm and increased thermal conductance by about 50% (Walsberg, Campbell & King 1978). At the same time, in still air the interception of incoming solar radiation increased by 30–100% depending on pelage colour (Walsberg, Campbell & King 1978; Walsberg 1988a). If capacity to flatten the pelage varies with size, this could have a large effect on the ability to dissipate heat. In the absence of information, we will assume that capacity to flatten the pelage is independent of body size (Calder 1984).

The complexities of heat transfer processes in fur and feathers have been subject of substantial previous modelling and empirical work (Cena & Monteith 1975a; Cena & Monteith 1975b; Walsberg, Campbell & King 1978; Walsberg 1988a; Walsberg 1988b). The bottom line of these models is that thermal conductivity of the pelage would be expected to increase as animals get larger, with a scaling exponent of approximately 0·17. Pulling all these mass dependencies together and assuming the surface temperature is equal to the ambient temperature (Ta), we can solve eqn 2 for the expected relationship between overall maximal capacity to dissipate heat (Hs) and Mb in the case of geometric similarity of the limbs. This gives


and consequently

image(eqn 5)

Eqn 5 suggests that Hs will scale in relation to Mb0·50 if the limbs scale geometrically (Hs will be proportional to Mb0·47 if the limbs scale elastically, as Mb0·66 on the numerator is replaced by Mb0·63). Maximal heat dissipation capacity will also be dependent on the difference between Tb and Ta. Consequently, animals in colder environments will be able to lose more heat. This analysis accords with several previous analyses of minimum heat loss in relation to body size in the region where all animals are below lower critical temperature (Peters 1983; Calder 1984; McNab 2002). In our case, the critical point is that all animals are below upper critical temperature, where thermal insulation is at a minimum, but the fundamental principles are the same.

Predicted responses of animals to the theoretical model of heat dissipation

The much lower scaling exponent for maximal capacity to dissipate heat (0·47–0·50) compared with the exponent for basal metabolic rate (BMR) (generally presumed in the range 0·69–0·76; Kleiber 1932, 1961; Zeuthen 1947; Hemmingsen 1960; Lasiewski & Dawson 1967; Stahl 1967; Dawson & Hulbert 1970; Calder 1974; White & Seymour 2003; Brown et al. 2004; McKechnie, Freckleton & Jetz 2006; Duncan, Forsyth & Hone 2007; Sieg et al. 2009; White, Blackburn & Seymour 2009) means that a fundamentally important prediction of the HDL theory is that scope to increase metabolism declines as animals get larger. Larger animals are therefore predicted to be under much greater constraint in their total energy budgets because they are unable to elevate their metabolism as high as smaller animals. Moreover, because the line for Hs in relation to Mb converges on the line relating BMR to Mb, these lines must ultimately meet at some point where Hs would equal BMR. Such an animal could not exist as it would have no scope to perform any activity above BMR. Indeed, a field metabolic rate (FMR) below about 1·3 × BMR is probably incompatible with sustained free-living existence (Goldberg et al. 1991). Our model therefore provides a metabolic basis for predicting the maximum possible body size for an endothermic animal. The MTE, on the other hand, predicts a scaling exponent for both BMR and FMR of 0·75 (Brown et al. 2004; West & Brown 2005), and hence the lines run parallel. A fundamental difference between the HDL theory and the MTE therefore is that the MTE generates no predicted constraint on scope to elevate metabolic rate as animals get larger, and no predicted maximal endothermic Mb.

We can make a number of predictions of how natural selection and evolution might respond to reduce the constraint imposed by the convergence of Hs and BMR in relation to body size. These predictions are as follows (note that because the prediction from the MTE is that the lines for BMR and FMR run parallel, the MTE does not predict any of these responses):

  • (a) Larger animals will reduce their contribution of Mb into the insulating shell (pp). Hence, instead of pelage mass being a constant proportion of total Mb, we predict a progressively reduced contribution as animals get larger, i.e. pelage mass proportional to Mb<1·0.
  • (b) Larger animals will evolve morphological features that aid heat dissipation. These would include, for example, disproportionately long limbs compared to limbs that conform to geometric similarity (leg length proportional to Mb0·33) or elastic similarity (leg length proportional to Mb0·25).
  • (c) Larger animals will increase their regulated Tb and thereby increase the driving gradient for heat loss.

The predictions (a) to (c) from the HDL model are all supported by empirical observations. In birds, plumage mass scales with an exponent of 0·95 (Turcek 1966), in mammals skin mass has an exponent of 0·942 (Pace, Rahlmann & Smith 1979) and fur mass has an exponent of 0·98 (data from Pitts & Bullard 1968; recalculated in Calder 1984). Limb lengths studied in bovids suggest that they increase as expected from elastic similarity (McMahon 1975b; Alexander 1977). However, data from a much broader sample of mammals ranging in size from shrews (Sorex) to elephant (Loxodonta) revealed scaling exponents between 0·33 and 0·38 (Alexander et al. 1979). In birds, femur length scales with an exponent of 0·36 (Prange, Anderson & Rahn 1979). These data suggest that large mammals and birds have longer legs than anticipated by either geometric or elastic similarity. Finally, although Calder (1984) suggested that Tb was independent of size in eutherian mammals and marsupials, White & Seymour (2003) using a much larger sample found that Tb in mammals scaled with Mb0·04, and the same exponent was observed in birds by Calder & King (1974). A more recent analysis suggests that across all mammals and birds there is no scaling relationship when some allowance was made for phylogenetic effects. This was, however, not a full phylogenetically independent contrasts (PIC) analysis. Moreover, within particular lineages there were significant relationships in both directions (Clarke & Rothery 2008).

There are two additional predictions from the HDL theory that remain untested. First, our model assumed that the core and shell components of the body are centred on the same mid-point. By offsetting the core relative to the shell, an animal makes the shell thinner at one side, but thicker at the other. The effects on heat loss are not cancelled out by this process. This can be easily appreciated if the effect is simplified by considering the situation where half of the sphere has the pelage depth halved, and the saved pelage is used to increase the thickness of the other half (Fig. 2). In the half where the pelage is thinned, the area is half of the total (0·5·A), and the depth is half what it was originally (0·5·d), so the animal now loses as much heat as it did originally across its entire surface, just over this thinned half (eqn 2). Since some heat will be lost through the thicker part, the net effect of this asymmetrical positioning of the core is to increase overall maximal heat flow. We would therefore predict that this asymmetry in core position would be more likely to evolve in larger animals. There are clear examples in larger animals with large asymmetries in fur thickness (Morrison 1966). This may also be driven by the need for larger animals to reduce uptake of solar radiation which would further compromise their heat dissipation capacity. The final prediction from the HDL theory is that larger animals should have greater capacity to flatten their pelage reducing its insulative value.

Figure 2.

 Symmetrical (a) and asymmetrical (b) location of the core relative to the shell and its implication for the depth of the shell (d) and the intensity of heat loss over the surface (wavy arrows). Symmetrical location of the core ensures equal distribution of heat loss through the surface, whereas asymmetry leads to variability in the shell depth and an associated negative correlation between the intensity of heat loss and d. A simplified model of this effect where the shell is halved in thickness over half the surface (c) shows that overall heat loss is increased when the core is asymmetrically located. In this model, the lower half of the animal has had a section of pelage removed (black), reducing the shell depth to 0·5·d, and transferred onto the top half, increasing the shell depth to 1·5·d (see text for further explanation).

Predictions from the model after incorporating empirical data

Using these empirical observations, we can modify the model predictions of scaling of maximal heat dissipation capacity (Hs) in relation to Mb. If pelage mass increases in relation to Mb with an exponent of 0·95–0·98, rather than 1·0, then thermal conductivity (k) will increase more rapidly than the derived scaling exponent of 0·17. Using a scaling of pelage mass to Mb of 0·97 suggests that inline image(i.e. exponent increased by 1·0 − 0·97 = 0·03) and pelage depth will scale as inline image (exponent decreased by 0·03). Similarly, if the limbs increase more rapidly in length than anticipated from geometric or elastic similarity, we can anticipate that surface area (A) will also increase more rapidly in relation to mass than Mb0·66. If we assume that inline image, then


and therefore

image(eqn 6)


image(eqn 7)

This model, taking into account the physical constraints on heat dissipation capacity and then accepting the evolutionary responses to these constraints in the key parameters, suggests that Hs should actually scale in relation to Mb with an exponent of 0·63. This is likely to be a minimum estimate because we have not taken into account the two additional predictions detailed above.

How does what we have done here differ from the accusations levelled at the proponents of the MTE that the fractal model uses empirical observations that inevitably yield the predicted scaling exponent of 0·75 (Kozłowski & Konarzewski 2004, 2005; Brown, West & Enquist 2005)? The difference is that we have built a model based entirely on physical principles of heat exchange and made explicit assumptions not based on empirical data. We have then generated a number of predictions of how evolution would likely respond to the consequences of such a model for scope to increase metabolic rate in larger animals. We have then compared these predictions with empirical data, and incorporated these predicted responses into the model. We acknowledge that this process of modification may be imperfect because there may be additional responses by large animals that we have not considered, and some that we have considered, but were unable to quantify because empirical data were unavailable. This means that the estimated scaling exponent of 0·63 is more likely to be an underestimate than an overestimate.

In addition to making a prediction of the scaling exponent (eqn 6), it is also feasible to solve the heat balance equations to obtain an exact prediction of maximal heat loss capacity for different sized animals at different ambient temperatures (details are in Supporting Information, Appendix S1). This quantitative model predicts, as anticipated from the mathematical treatment above, a linear increase in log converted Hs in relation to log converted Mb. This model quantification revealed no interaction of the predicted effects of Mb and Ta on Hs (i.e. the lines relating maximal heat loss to Mb run parallel at different temperatures). These predictions are featured in Figs 3a and 4 in addition to empirical measurements to test the model.

Figure 3.

 (a) Effect of body mass (Mb) on the field metabolic rate (FMR) in mammals measured by the doubly-labelled water method. Data are 290 measurements across 123 terrestrial species. Overlaid on the empirical data are the quantitative predictions for maximal heat loss capacity at ambient temperatures 30°C and 10°C. (b) Residual variation in FMR of mammals with Mb effect removed plotted against ambient temperature.

Figure 4.

 Effect of body mass (Mb) on the field metabolic rate (FMR) in birds measured by the doubly-labelled water method. Data are 130 measurements on 130 species. Overlaid on the empirical data are the quantitative predictions for maximal heat loss capacity at ambient temperatures 30°C and 10°C.

Testing between the MTE and HDL theories: field metabolic rate

Heat is generated as a by-product of all metabolic processes. It is generated at different rates depending on what the animal is doing. In endotherms, heat production is low when the animals are in the basal state (BMR). That is when the non-reproductive animal is at rest, within the thermoneutral zone, maintaining Tb at euthermic levels, and in a post-absorptive state (Kleiber 1932, 1961). Even lower levels of expenditure can occur if the animal relaxes regulation of Tb and allows its body to cool, as observed during hibernation or daily torpor (Heldmaier & Klingenspor 2000; Rikke et al. 2003; Schubert et al. 2010). Physical activity, digestion and temperatures below the lower or above the upper critical temperatures all increase energy demands above BMR. Very high rates of heat production correspond with a state of maximal oxygen consumption called VO2max (e.g. Taylor et al. 1981; Hinds et al. 1993; Bishop 1999; Weibel et al. 2004). This may be elicited either by exceptional physical activity or by extreme cold. Generally, the time averaged metabolic rates (= heat production) over periods of a day or so are called the average daily metabolic rate (ADMR) or daily energy expenditure (DEE). If ADMR or DEE are measured in the field, it is also called FMR.

An important question is what level of energy expenditure is most appropriate to compare with the maximal heat dissipation capacity (Hs) derived from the HDL theory, and the maximal capacity to distribute resources predicted by the MTE? Most previous studies testing the MTE in endotherms have focussed on the scaling exponent with Mb at the level of BMR (e.g. Savage et al. 2004; Farrell-Gray & Gotelli 2005; Duncan, Forsyth & Hone 2007; Sieg et al. 2009; White, Blackburn & Seymour 2009). This seems, however, to be an inappropriate level to evaluate whether the fractal distribution network limits the rate of metabolism. While BMR is a repeatable and broadly comparable index of metabolism across species, in the basal state animals cannot be operating at their maximum limits. It seems unlikely, therefore, that the exponent of around 0·75 in the scaling of BMR against Mb comes about because of the limits imposed by a fractal resource distribution network.

One might imagine then that the most relevant comparison to the predictions of the HDL and MTE theories would be VO2max. However, we suggest that this is not the case. First, VO2max is generally measured under unusual conditions that radically increase the capacity of animals to dissipate heat. For example, animals may be exposed to gas mixtures of helium and oxygen (e.g. Chappell, Bachman & Odell 1995; Gębczyński & Konarzewski 2009) or forced to swim in cold water (e.g. Konarzewski, Sadowski & Jóźwik 1997; Sadowska et al. 2005). Second, we suggest that the level of energy expenditure, and hence heat production, attained under these conditions has been moulded primarily by ecological events where avoidance of overheating is probably of minor significance. In particular, we suggest that VO2max when measured during maximal physical exertion probably reflects the evolutionary response to avoid predation. When an animal is being chased by a predator, the long-term negative fitness consequences of hyperthermia are greatly outweighed by the immediate fitness benefits of not being eaten. Hence, maximal performance is unlikely to be bounded by the considerations detailed above concerning Hs. The value of VO2max will certainly exceed Hs, and therefore commonly result in temporary hyperthermia (or not, if capacity to dissipate heat is artificially elevated such as by simultaneous immersion into cold water). We suggest, instead, that the most relevant comparator to the predictions of the HDL and MTE theories is the FMR. This is because the time averaged heat production over days rather than minutes is directly related to fitness in terms of reproductive output, but also because the FMR is potentially sensitive to the longer term negative consequences of hyperthermia for survival, via the mechanisms outlined above. We suggest, similarly, that the most appropriate metric for testing the MTE is also FMR because this is where fractal limitations in the capacity to distribute resources (nutrients) will be felt most acutely on fitness (i.e. reproductive output and survival).

To test between these contrasting models based on the HDL theory and MTE, we compiled information on FMR measured in free-living terrestrial mammals and birds. Measurement of DEE in free-living animals is now routine and can be made using several different approaches, the main ones being the doubly-labelled water (DLW) technique (Lifson & McClintock 1966; Speakman 1997) and heart rate telemetry (Butler et al. 2004). The costs and benefits of the different approaches have been previously reviewed (Butler et al. 2004). In practice, evaluations of FMR available from heart rate measurements are restricted to relatively few larger species, while the DLW method has now been applied to over 250 species of mammals and birds (previous reviews in Nagy, Girard & Brown 1999; Speakman 2000; Anderson & Jetz 2005; Nagy 2005). We have recently compiled data on the FMRs measured by the DLW method in both mammals (Westerterp & Speakman 2008) and birds (Furness & Speakman 2008), published in the context of obesity and ageing. For the current paper, we updated the database for mammals taking into account publications up to June 2009 and some of our own more recent studies.

In total, we found 290 measurements of FMR for 123 terrestrial mammal species covering a size range from the 8·0 g pipistrelle bat Pipistrellus pipistrellus, (Racey, Speakman & Swift 1987) to the red deer Cervus elaphus, (Haggarty et al. 1998) weighing 107·3 kg. Typical examples of field studies of metabolic rate in mammals include Scantlebury et al. (2006) and Lane et al. (2010). For the mammal data, we also extracted information on the ambient temperature at which the measurements had been made, and separated data collected under different conditions – hence the number of measurements exceeds the number of species. In total, Ta data were available for 227 of the 290 measurements. We also found 130 measurements across 130 species of birds covering a size range from 3·3 g in the black-chinned hummingbird Archilochus alexandri (Powers & Conley 1994) to 78·5 kg in the ostrich Struthio camelus (Williams et al. 1993). Typical examples of field studies of metabolic rate in birds include Bryant & Westerterp (1983) and Davis, Croxall & O’Connell (1989).

We treated FMR data measured under different temperature conditions in mammals as independent measurements and log transformed them to derive the relevant scaling exponents. We then performed a multiple regression on the resultant data with LogeMb and Ta as independent predictors for Loge FMR. The effects of both LogeMb and Ta were highly significant (Fig. 3a and b). There was no significant LogeMb by Ta interaction (LogeMb, F1,223 = 394·5, < 0·001; Ta, F1,223 = 14·74, < 0·001; Loge Mb · Ta, F1,223 = 0·14, = 0·714), consistent with the quantification of the model (Supporting Information, Appendix S1). The resultant least squares regression


explained 92·3% of the variation in Loge FMR. The scaling exponent of 0·647 in relation to LogeMb had a SE of 0·0127. This exponent did not differ significantly from the exponent of 0·63 predicted by the HDL theory (= 0·017/0·0127 = 1·34, > 0·05) but did differ significantly from the exponent of 0·75 predicted by the MTE (= 0·103/0·0127 = 8·11, < 0·001) (note that we adopt here the classic exponent of 0·75 predicted by the MTE and not the more recent suggestion that the MTE predicts an exponent of 0·81; Savage, Deeds & Fontana 2008). Excluding Ta as a factor increased the available measures to 290. In this analysis, the least squares regression was


The scaling exponent in this case increased to 0·665 (SE = 0·0115, r2 = 0·920), which differed significantly (< 0·01) from the predictions of both HDL and MTE models. Condensing the data to only a single average datum for each species yielded the least squares regression


Again, the scaling exponent of 0·670 (SE = 0·0156, r2 = 0·938) differed significantly from both theoretical predictions.

Overlaid on Fig. 3a are the quantitative predictions of maximal heat loss capacity at 10 and 30°C derived from the model presented in Supporting Information, Appendix S1. It is important that these lines overlap with much of the empirical data. This is because there might be a good match between the observed and predicted scaling exponents, but the actual HDL constraint might be located higher than the level at which most animals expend energy. Such a pattern would be consistent with expenditure being extrinsically constrained, or constrained by a different intrinsic factor. The overlap of the theoretical and empirical data supports our hypothesis that in many situations mammals will be under maximal heat dissipation constraints.

In the birds, the least squares regression equation


explained 92% of the variation in FMR (Fig. 4). The standard error for the scaling exponent was 0·0172 and hence this also did not differ significantly from the predicted exponent from the HDL theory (= 0·028/0·0172 = 1·63, > 0·05) but did differ significantly from the MTE (= 0·092/0·0172 = 5·35, P < 0·001). We also made quantitative predictions of maximal heat loss capacity for birds at 30 and 10°C and these predictions are overlaid on Fig. 4. As with the mammals, many of the bird data overlap with the theoretical predicted maximal heat loss capacities. The mean species data for both birds and mammals are available as Appendices S2 and S3 in Supporting Information, and also at the DLW resource centre (– note British spelling of ‘labelled’).

It has recently been suggested that the relationship between BMR and Mb does not conform to a simple power law and is better suited to a quadratic model with convex curvature (Clarke, Rothery & Isaac 2010; Kolokotrones et al. 2010; White 2010). Visual examination of Fig. 3a may lead to the impression that Loge FMR is also curvilinearly related to LogeMb. However, in this case, the higher metabolic rates in the larger animals, and hence their higher residuals, are explained by the fact that these larger animals are generally measured at lower ambient temperature (Fig. 3b). We formally examined the possibility that the FMR data summarized here also do not conform to a simple power relation by examining the relationship between the residuals and fits of the models, including both LogeMb and Ta as predictors for mammals and Loge Mb as a predictor for birds. In both mammals and birds, there was no indication of a systematic deviation in the residuals from a simple power law, indicating that Loge FMR is not curvilinearly related to LogeMb.

These conventional analyses may be compromised by lack of phylogenetic independence (Garland, Harvey & Ives 1992; Garland et al. 1993; Garland, Bennett & Rezende 2005). To overcome this issue, we pooled data within species and then calculated the PIC for both FMR and Mb across all 123 species of terrestrial mammal and 130 species of bird, using the high level phylogenies for mammals (Murphy et al. 2001) and birds (Sibley & Ahlquist 1990), supplemented by additional sources for individual groups. Using PIC, the gradient of the relationship between Loge FMR and LogeMb in mammals was 0·679 (SE = 0·032; comparison to the HDL theory, = 1·53, > 0·05; comparison to the MTE, =2·22, < 0·05). In birds, the gradient of the relationship using PIC was 0·576 (SE = 0·036; comparison to the HDL theory, = 1·50, > 0·05; comparison to the MTE, = 4·97, < 0·001) (plots of PIC corrected data are presented in Supporting Information, Appendix S4).

Overall, the data did not exactly fit the HDL theory prediction of a scaling exponent of 0·63, but the observed conventional exponents of 0·647 for mammals and 0·658 for birds, and the PIC exponents of 0·679 and 0·576 for the two classes, were much closer to and not significantly different from the HDL theory, but all differed significantly from the MTE prediction (0·75). Residual variation, once the effect of Mb was accounted for, was negatively related to Ta as anticipated, because Ta should influence heat dissipation capacity (eqn 7). Most importantly, the quantitative prediction of maximal heat loss capacity and observed expenditures overlapped in both data sets. These data therefore provide substantially more support for the HDL theory than the MTE model.

Additional predictions and consequences of the HDL theory

Energy demands of aquatic mammals and maximal body size of endotherms

Water has 23 × greater thermal conductivity than air (Schmidt-Nielsen 1975). Consequently, the HDL theory predicts that large aquatic mammals should be far less constrained in their FMR than terrestrial mammals because their maximal capacity to dissipate heat (Hs) will be correspondingly greater. The HDL theory predicts that their DEEs should be able to break out above the constrained levels for large terrestrial animals. We found in the literature an additional 22 measurements of FMR for ten species of aquatic mammal made using the DLW technique. Pooling the data for both terrestrial and aquatic mammals, the FMRs in aquatic mammals (Fig. 5) were significantly higher than observed for the terrestrial mammals (F1,309 = 7·97, < 0·001). It had been noted previously that seals appear to have exceptionally high FMR and suggested that this disparity between aquatic and terrestrial mammals perhaps reflected a methodological problem with the DLW technique (Boyd et al. 1995). However, a recent validation study in pinnipeds shows that the method does work as anticipated in these animals (Sparling et al. 2008). The high metabolic rates of aquatic mammals are consistent with the prediction that in the aquatic habitat their Hs is enormously elevated, releasing them from the constraint imposed on equivalent sized terrestrial animals. By contrast, the MTE makes no prediction for such an elevation in FMR of pinnipeds as there is no a priori reason to believe that this particular order of mammals has a unique fractal distribution system that is substantially more efficient.

Figure 5.

 Field metabolic rates (FMR) of aquatic mammals and terrestrial mammals weighing more than 3 kg (LogeMb = 8·0) with the least squares regression lines fitted. Aquatic mammals have significantly elevated FMR relative to terrestrial mammals as predicted from the HDL theory based on the greater thermal conductivity of the aquatic medium in which they live. The statistical analysis referred to in the text includes all measurements for both aquatic and terrestrial mammals.

Above we introduced the notion that the convergence of the relationship between Hs and BMR would ultimately constrain the maximum possible size for an endotherm. The fact that aquatic habitats permit greater heat dissipation leads to the prediction from the HDL theory that the largest possible endotherm living in water will be much larger than the largest possible endotherm living on land. Considering extant endotherms, this is undoubtedly the case. The blue whale Balaneoptera musculus with an Mb of about 80–100 × 106 g weighs substantially more than the African elephant Loxodonta africanus with an Mb of at most 7–9 × 106 g. However, the fossil record reveals many extinct terrestrial animals that were much larger than modern elephants. The largest of these, the sauropod dinosaur Argentinosaurus, approaches in predicted mass that of the blue whale (Mazzetta, Christiansen & Fariña 2004). However, this comparison is potentially compromised in several ways. First is the accuracy of the predicted mass of this animal from relatively little fossil material (Paul 1997). Second, the possibility that its lower body was routinely immersed in water allowing increased heat dissipation, and finally, the disputed thermoregulatory status of the dinosaurs may also compromise the comparison (Bakker 1972; Ruben et al. 1996; Seebacher 2003). The largest unequivocally terrestrial mammal (and by phylogenetic implication certainly endothermic) was probably Baluchitherium, which appears to have had an Mb of about 15–20 × 106 g, substantially larger than the modern African elephant, but as predicted from the HDL theory – much smaller than the blue whale. Predictions of the exact size limits for mammals in air and water should be possible, providing an additional method to test the HDL model.

Adaptations to enhance heat dissipation capacity

A major feature of our theory is that endothermic animals oscillate between situations and times when energy supply is limited (for example winter in the temperate and arctic zones, e.g. Speakman et al. 2003; Humphries et al. 2005; Zub et al. 2009) and situations or times when energy is unlimited (e.g. summer in the same regions). During the periods that energy supply is limited, endotherms are expected to evolve traits that minimise energy demands, allowing them to match their requirements to the external availability of energy. One such feature is the external insulating pelage. This serves to retard heat loss by lowering the rate of conduction of heat from the body core to the surface. When energy is limited, a dense and deep pelage that reduces heat loss is an advantage for any animals living at temperatures below their upper critical temperature. If energy supply was always limited, as supply side ecology (including the MTE) assumes, we would predict that animals should always retain a maximally insulating pelage, because this would minimise their thermoregulatory demands. However, a thick insulating pelage is a disadvantage in situations where energy supply is unlimited and expenditure is constrained by capacity to dissipate body heat. This is because the pelage insulation becomes the primary constraint on heat loss. The HDL theory therefore predicts that animals should seasonally moult their pelage between a thick insulating layer that retards heat loss when energy supply is limited, to a thin layer that promotes heat dissipation while still retaining the other functions that the pelage serves (e.g. protection from ectoparasites and signalling, Stettenheim 2000). Moulting the pelage is a pervasive feature of both mammal and bird life cycles and appears energetically costly to perform (e.g. Lindstrom, Visser & Daan 1993; Schieltz & Murphy 1997; McNab 2002). Replacing worn feathers and fur may be necessary to maintain pelage integrity, so the fact that animals do this does not necessarily support the predictions of the HDL theory. The key prediction is whether the insulating value of the pelage is different between seasons. Substantial evidence accumulating from at least the 1950s suggests that this is the case (Scholander et al. 1950). For example, Fig. 6 shows the energy expenditure of the willow ptarmigan Lagopus lagopus as a function of ambient temperature (Ta) in summer and winter (Ricklefs 1983; after West 1972). The gradient of the relationship between energy expenditure and Ta below thermoneutral zone reflects the insulation of the pelage. Ptarmigan in winter have a more insulating pelage than in summer. The classic adaptive interpretation of moulting into a winter pelage is that the winter coat reduces demand for energy in the face of declining Ta at a time when resources may be particularly scarce. A ptarmigan at -30°C in winter reduces its energy demands by about 75% by moulting into the winter coat (Fig. 6). The question that is never asked is if energy resources are always environmentally limited, or limited by a supply side fractal network, why does the animal ever take this winter coat off? Put another way, in a resource limited environment what is the advantage of moulting into a pelage that increases thermoregulatory demands? A ptarmigan in summer, for example, moults into a pelage that at 0°C increases its energy needs by 30% (Fig. 6).

Figure 6.

 Effects of season on thermal insulation of the pelage of the willow ptarmigan Lagopus lagopus (from Ricklefs 1983; after West 1972). In summer, the gradient of the relation between metabolic rate and temperature is steeper than in winter.

The HDL theory provides an answer to this question. Animals take off their winter coats to maximise their capacity to dissipate heat in a situation where environmental resource supplies are unlimited. Specifically, the HDL model predicts that lactating females will benefit from pelage thinning in the breeding season more than males and therefore are predicted to have less insulating fur. We are unaware of existing data to test this prediction, but they should be easy to collect. The HDL theory also predicts that females would have morphological features that promote heat loss relative to males. This would include longer limbs. Relatively longer limbs in humans are associated with higher resting metabolism reflecting greater heat loss (Tilkens et al. 2007). It is also well established that female humans have relatively longer legs than males (Swami, Einon & Furnham 2006). Relatively long legs in females are regarded as physically attractive by males (at least in western cultures), while shorter legs in males are normally sexually attractive to females (but not always, see Sorokowski & Pawłowski 2008). The selection pressures on leg length in females, and why they are considered attractive by males, have generally revolved around the suggestion that relatively long legs may be correlated with wider pelvises which allow easier births and higher birth weight, both of which reduce infant and maternal mortality (Martorell et al. 1982; Sokal, Sawadogo & Adjibade 1991). It is possible, however, that because longer legs promote heat dissipation, they facilitate higher milk production and hence better offspring growth and reduced infant mortality. Fielding et al. (2008) found that leg length in a cohort of over 9000 women from China was associated with greater numbers of offspring, but there was no such association in males. In other mammals, sexual dimorphism in limb lengths, their role as sexual signals and their link to reproductive performance is less clear (e.g. Taylor 1990), but should be highly amenable to direct testing. Similarly, in birds because contributions to rearing offspring are less sexually dimorphic and female birds do not lactate, we would not anticipate large sexual dimorphism in leg lengths relative to body length. This is also amenable to direct testing.

Another way that animals reduce their energy demands when energy supplies are limited is to modify their environment by constructing a nest. Nests provide insulation that the animal does not need to carry around all the time, and they allow the animal to build up a thermal microclimate inside that reduces energy demands (e.g. Hayes, Speakman & Racey 1992; Dechmann, Kalko & Kerth 2004). In fact, elevated nest temperature relative to outside is a standard diagnostic tool to infer nest attendance (e.g. Schneider & McWilliams 2007). Nests also provide a convenient refuge in which to hide and house offspring. If energy supply was always limited, we might expect nest construction to be relatively constant and designed to maximise the retention of heat, thereby minimising thermoregulatory demands while the animals are in residence. The HDL theory predicts that female mammals raising offspring will benefit from building nests that serve the function of a refuge for their offspring and with sufficient insulation to minimise thermoregulatory demands of the offspring thereby facilitating their growth, but not be so insulating as to retard her capacity to effectively dissipate lactogenic heat. Two studies support these predictions. In the field vole Microtus agrestis, male voles build better insulated nests than females (Redman, Selman & Speakman 1999). Furthermore, female red squirrels supporting large litter masses of furred offspring, later in the summer when ambient temperatures are generally warmer, occupied nests of lower insulative value than females raising litters of furless pups when temperature norms were cooler (Guillemette et al. 2009). The generality of these findings regarding nest insulation are uncertain but amenable to further investigation.

Bergmann’s rule

In the late 1800s, the German naturalist Carl Bergmann observed that individuals in mammalian species collected from higher latitude regions had larger body masses than individuals of the same species collected from lower latitudes. His interpretation of these data (along with many subsequent interpretations) was that the ‘favourable’ low surface to volume ratio of the larger animals permitted them to conserve heat better in colder higher latitude regions, reducing their energy demands. The fallacy of this argument has been repeatedly emphasised (e.g. McNab 1971, 2002; Steudel, Porter & Sher 1994). While energy demands per gram of larger individuals is reduced, their total energy expenditures are higher, and animals must eat food to cover their total energy requirements rather than their mass-specific demands. However, it has been noted that while total energy demands increase with body size, the ‘favourable’ low surface to volume ratio means that the rate of this increase is much slower than the rate at which fat storage increases with body size, particularly in cold conditions (Calder 1984). In a situation where there is no external food and the animal relies completely on its stored reserves, the ‘fasting endurance’ of the larger animals will be greater (Calder 1984; Millar & Hickling 1990; Mugaas & Seidensticker 1993). This advantage will pertain at all latitudes but its effects on survival and selection will be more intense at higher latitude.

Although there are many exceptions to Bergmann’s rule and also many counter examples (e.g. Geist 1987; Medina, Marti & Bidau 2007; Olson et al. 2009), the fasting endurance hypothesis remains a potentially valid interpretation for those species where it is observed. The diversity in whether a species conforms or not to Bergmann’s rule may also be because of the differing importance of fasting endurance in the individual species’ ecology. Similar effects of altitude have also been observed (e.g. Landmann & Winding 1995; Wigginton & Dobson 1999) and similarly interpreted as reflecting the favourable low surface to volume ratio of larger individuals in colder high altitude conditions for fasting endurance.

Bergmann’s and subsequent interpretations that the lower surface to volume ratio of larger individuals is ‘favourable’, depends on the pre-conceived notion that it is cold exposure that is the ecological problem to be solved by adaptation in body size. The HDL theory provides an alternative viewpoint. If heat dissipation during reproduction is a key issue, then it is actually the smaller individuals in the hotter lower latitudes that have a more ‘favourable’high surface to volume ratio which promotes heat dissipation giving them a greater reproductive success than their larger conspecifics. This advantage of being small in terms of reproductive success pertains at all temperatures. Its effects, however, will be proportionately lower as it gets cooler because the driving gradient for heat loss becomes greater anyway as it gets colder (see Fig. 3a). Selection favouring smaller body size will be more intense therefore at lower latitudes and altitudes that are warmer.

Clearly, in mammals the HDL theory predicts that more selective pressure to reduce body size in warmer conditions will be felt in females. This suggests that sexual size dimorphism should also vary in relation to latitude, with dimorphism being greater at lower latitudes or where it is warmer. Supporting this prediction, it has been previously observed that male and female body size respond differently to environmental variation in strongly dimorphic mammals. For example, in red deer Cervus elaphus living in northern Norway, progressive warming of the climate over the period from 1957 to 1996 led to an increase in the size of males, but a decrease in the size of females (Post et al. 1999). Contrasting these observations, however, sexual size dimorphism in human populations appears to peak at around 40 absolute latitudinal degress (Gustafsson & Lindenfors 2009).

Basal metabolic rate

A number of comparative studies of BMR in mammals and birds have demonstrated that in addition to a large effect of Mb, there is an additional independent effect of either ambient (habitat) temperature or latitude, with animals from warmer climates (or lower latitudes) having reduced BMR once the effects of mass are taken into account (e.g. Speakman 2000; Lovegrove 2003; Rezende, Bozinovic & Garland 2004; Jetz, Freckleton & McKechnie 2007; White et al. 2007). The fact that most animals appear to be under heat dissipation constraints acting to limit FMR provides a potential explanation for this phenomenon. Specifically, by reducing BMR under hot conditions, when FMR is most constrained, animals increase their potential metabolic scope (FMR/BMR) and their absolute capacity to invest energy in activity or reproduction (FMR – BMR). This recalls the suggestion of McNab & Morrison (1963) that low BMR in hot environments might be an adaptation to reduce endogenous heat production. The HDL theory provides a context for understanding the significance of this reduced BMR.

Potential problems with the HDL theory

Faced with the HDL theory, two immediate and very reasonable objections have been raised:

  • 1 Animals routinely exercise and when they do so, their metabolic rates elevate enormously but they are somehow able to dissipate the resulting body heat. Moreover, some migrating birds may fly continuously for over 24 h. How can it be that the maximum ability to dissipate heat (Hs) is lower than the active metabolic rate which is around 10–15 × BMR?
  • 2 Why do breeding females at peak lactation or birds feeding nestlings not shed their pelage completely?

These potential problems with the HDL theory primarily arise because the model we have developed concerns Hs of a stationary animal. When animals exercise, they generally move through the environment increasing heat flow because of forced convection at their body surface. Heat loss by increased convection accounts for 80% of the total heat loss in a flying bird (Ward et al. 1999, 2004). For diurnal animals, this elevated heat loss will be offset by an increase in incoming solar radiation. Nevertheless, because non-exercising animals are stationary, their maximum capacity to dissipate heat is likely to be lower than for exercising animals. By taking into account movement, the Hs generated by the quantitative model (Supporting Information, Appendix S1) and represented by the theoretical lines in Figs 3a and 4, would rise slightly.

Even with this extra capacity to lose heat, running mammals and flying birds face problems dissipating the heat generated from their exercise and have evolved mechanisms to eliminate the extra heat load that would not be sustainable over protracted periods. These mechanisms include temporary hyperthermia (Butler, West & Jones 1977), thereby elevating the driving gradient for heat loss by convection and radiation, and increasing evaporative water loss. Increased evaporation accounts for about 11% of heat loss during flight (Ward et al. 2004). We have already detailed the negative impact of hyperthermia. Limitations on the duration that both these mechanisms can be utilised have been suggested to underpin limitations in exercise performance (e.g. Cheuvront & Haymes 2001; Hargreaves 2008; Simmons, Mundel & Jones 2008). The case of migrating birds is particularly instructive concerning the importance of limitations in capacity to dissipate heat in this respect. Most migrating birds fly at high altitude and often at night (Klaassen 1996; Léger & Larochelle 2006). These behaviours mean that the animals are not exposed to solar radiation and they experience much cooler ambient temperatures, facilitating heat loss without the need to increase evaporation or Tb (Portugal et al. 2009).

Regarding the complete loss of pelage, it is perhaps important to recognise that the fur and feathers play roles other than thermal insulation, including for example signalling (Stettenheim 2000), which may make their complete loss disadvantageous. Setting aside this possibility, however, it is clear that even when the environmental supply of energy is not limiting, there may be advantages to minimising the time spent collecting that energy. Because energy is available outside the nest and Ta outside is generally lower than inside, foraging animals will face a thermoregulatory requirement whenever they feed. They must satisfy this thermoregulatory demand before they can allocate collected energy to other activities like reproduction. If an animal minimises this thermoregulatory demand by having an insulating pelage, it will be able to devote more of the energy it collects to reproduction, but investment in reproduction will be capped by its capacity to dissipate heat when inside the nest. Alternatively, if the animal sheds all its pelage, it will have a greater capacity to allocate energy to reproduction, but it will also have greatly elevated demands when foraging. Consequently, much more of the energy the animal collects will have to be allocated to the thermoregulatory demands during the time it spends foraging. An additional potential factor may be the risk of predation when out foraging, which may place a premium on minimising foraging time. The optimum pelage depth therefore is a trade-off between heat conservation when outside the nest and heat dissipation when inside the nest (Fig. 7). It is unlikely that this optimum will sit at complete nakedness, but as indicated above, the optimum will likely be more towards the naked end of the spectrum in lactating females compared with males.

Figure 7.

 Optimisation of pelage depth to maximise energy allocation to reproduction. The maximum capacity to dissipate heat declines as the pelage gets thicker. However, the thermoregulatory cost of foraging also declines as the pelage depth increases. Consequently, the net energy gain from foraging (dashed line) increases as the pelage gets thicker because less energy is wasted on thermoregulation when foraging. At a certain point, the net gain from foraging crosses the maximum heat dissipation line. Further energy may be available in the environment but it is unusable because the animal cannot dissipate the heat associated with processing the extra food. Consequently, where the gain line crosses the maximum dissipation line defines the optimal pelage depth (dotted arrow).

By this model, if a female does not leave the nest to feed or a situation pertains where Ta (and other heat loss related parameters such as wind speed and solar radiation) inside the nest are equal to that outside, she might optimise investment by shedding all her pelage. Perhaps this is the situation in the naked mole rat Heterocephalus glaber, which spends almost its entire life underground, and a breeding ‘queen’ encased in a nest chamber she never leaves is supplied with food by members of a worker caste (Lacey & Sherman 1991). More intriguingly, however, we might speculate in mammals a scenario where there is selection for a division of labour between the sexes. The female may maximise her potential investment in offspring by largely shedding her pelage to facilitate lactation performance, while the male supplements her food intake thereby minimising her need to leave the nest and forage for herself. We are not aware of any examples of such a strategy, however, paternal presence appears to influence reproductive performance in Djungarian hamsters Phodopus campbelli by improving their capacity to cope with hyperthermia by some unknown mechanism (Walton & Wynne-Edwards 1998). This may conceivably involve the female reducing pelage insulation when males are present. Another candidate is obviously early hominids, providing an additional hypothesis for nakedness in ‘the naked ape’ (Morris 1967).


Our fundamental, some might say heretical, suggestion in this paper is that endotherms are constrained in the maximal level of energy they can expend by their capacity to dissipate internally generated heat to avoid hyperthermia. We suggest that this HDL constraint is a key factor that plays a role in many features of endotherm ecology, some important examples including the patterns of scaling in FMR and some other implications are elaborated above. These examples serve to emphasise the pervasive nature of this limitation. There are many additional implications and predictions of this theory that we do not have space to elaborate here. The importance of the HDL on endotherm ecology has been previously neglected in favour of models that emphasise extrinsic environmental limitation in energy supply or intrinsic limits in the resource supply distribution network (the MTE). Direct comparison of the predictions from the HDL theory and MTE for the scaling of FMR made here strongly favours the HDL theory. This leads us to suggest that energy supply may commonly not be a limited resource for endotherms, and that endothermic animals often therefore do not need to trade-off energy between competing demands because of a paucity of supply – extrinsically, alimentary or fractally mediated. However, they do need to trade-off competing processes that generate heat within the overall capacity for heat dissipation. Our analysis indicates that the HDL is of primary importance in endotherms. For ectotherms, it is likely that heat dissipation constraints are not important as these animals do not generate significant levels of heat internally. However, HDL in endotherms only assumes importance in the context of avoiding hyperthermia, and this may be an equally potent selective force in ectotherms, but mediated by different factors. One thing is certain, if energy supply is frequently unlimited for endotherms, it seems likely that it is often also unlimited for ectotherms.

Our model is novel because it shifts the emphasis away from the supply side of the energy balance equation and the resultant competition to allocate limited resources between competing demands (Fisher 1930; Gadgil & Bossert 1970), which hitherto has been considered the primary (or only) role played by energy in ecological systems. Rather, we emphasise the heat generating consequences of different morphological, physiological and behavioural strategies and how these traits compete within a boundary defined by the maximal capacity to lose heat. This altered viewpoint provides many insights into diverse aspects of mammalian and avian ecology. We suggest that an appreciation of the important role of limitations on energy expenditure, mediated via the HDL, will enhance our understanding in many additional areas. For example, as global climate issues become increasingly important, predicting how animals will respond to these changes assumes an even more important role. The classical energy allocation model suggests that the main impact of Ta changes will be on primary productivity and therefore indirectly on animals via energy supply. The HDL theory suggests, however, that changed temperatures will alter the position of the HDL, altering the constraint on energy expenditure. Impacts of climate changes on endotherms will probably therefore be more direct than has previously been appreciated. Of particular importance will be the effects of periods of high Ta and elevated hyperthermia risk. These are already known to exert a tremendous effect on the dairy industry, including reduced yields and direct mortality of lactating animals (e.g. West 2003; Stull et al. 2008). We suggest that inclusion of these effects into models attempting to predict effects of environmental changes on wild animals (e.g. Thomas et al. 2001; Humphries, Thomas & Speakman 2002; Root et al. 2003) may enhance the predictability of these models.


Our work on energetics of lactation that led to the heat dissipation limit theory has been funded by the UK Biotechnology and Biological Sciences Research Council (BBSRC grant BB/C504794/1), Natural Environmental Research Council (NERC grant NE/C004159/1) and the Royal Society (International Joint Project JP071172). We are grateful to the numerous honours, Erasmus exchange, masters and PhD students and postdocs that have worked on these projects. We thank Lindsey Furness who performed the PIC analysis on the bird data. Don Thomas, Henk Visser, Simon Verhulst, Serge Daan, Lobke Vaanholt, Mirre Simmons, Menno Gerkma, Colin Selman, Kim Hammond, Teresa Valencak, Thomas Ruf, Jan Kozłowski, Paweł Koteja, Marek Konarzewski, Karol Zub, Zhijun Zhao and Dehua Wang all made useful contributions to discussions around these ideas. We thank Peter Thomson, Brian Stewart and the staff of the animal house at the University of Aberdeen for their vital technical support. Two referees who waived anonymity (Murray Humphries and Craig White) made extensive and invaluable comments on earlier drafts of the manuscript that significantly improved it, and for which we are extremely grateful.